A014738 Squares of even triangular numbers.
36, 100, 784, 1296, 4356, 6084, 14400, 18496, 36100, 44100, 76176, 90000, 142884, 164836, 246016, 278784, 396900, 443556, 608400, 672400, 894916, 980100, 1272384, 1382976, 1758276, 1898884, 2371600, 2547216, 3132900, 3348900
Offset: 0
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).
Programs
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GAP
List([1..30], n-> ((2*n+1)*(2*n+1-(-1)^n))^2/4); # G. C. Greubel, Jul 24 2019
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Magma
[((2*n+1)*(2*n+1-(-1)^n))^2/4: n in [1..30]]; // G. C. Greubel, Jul 24 2019
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Mathematica
Select[Accumulate[Range[100]],EvenQ]^2 (* Harvey P. Dale, Oct 09 2012 *)
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PARI
vector(30, n, ((2*n+1)*(2*n+1-(-1)^n))^2/4) \\ G. C. Greubel, Jul 24 2019
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Sage
[((2*n+1)*(2*n+1-(-1)^n))^2/4 for n in (1..30)] # G. C. Greubel, Jul 24 2019
Formula
a(n) = A014494(n + 1)^2. - Sean A. Irvine, Nov 18 2018
From G. C. Greubel, Jul 24 2019: (Start)
G.f.: 4*x*(9 +16*x +135*x^2 +64*x^3 +135*x^4 +16*x^5 +9*x^6)/((1-x)^5*(1+x)^4).
E.g.f.: x*(35+41*x+36*x^2+4*x^3)*cosh(x) + (1+9*x+77*x^2+28*x^3+4*x^4)* sinh(x). (End)
From Amiram Eldar, Mar 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 7*Pi^2/12 + 2*Pi - 12.
Sum_{n>=0} (-1)^n/a(n) = 12 - 4*G - 12*log(2), where G is Catalan's constant (A006752). (End)
Extensions
More terms from James Sellers